Semicircular contour theorem

Statement

Version where the integral over the contour tends to zero

Suppose $f$ is a meromorphic function on an open subset of $\mathbb {C}$ containing the closed upper half-plane. Further, suppose there exists $k>1$ and a constant $M$ such that for all sufficiently large $R>0$, we have:

$|f(z)|\leq {\frac {M}{|z|^{k}}}$

for $z$ in the upper half-plane. Then, if $\gamma _{R}$ denotes the semicircular arc of radius $R$ centered at zero, then:

$\lim _{R\to \infty }\left|\int _{\gamma _{R}}f(z)\,dz\right|=0$

In particular, if $f$ has no poles on the real axis, and it has finitely many poles $z_{1},z_{2},\ldots ,z_{n}$ in the upper half-plane, we get:

$\operatorname {PV} \int _{-\infty }^{\infty }f(x)\,dx=2\pi i\sum _{j}\operatorname {res} (f;z_{j})$